Standard state infinitely dilute solution

We note that the standard state, infinite-dilution chemical potential, involves only solute-solvent interactions, since the coupling energy in the Boltzmann factor in Eq (3.61) is computed with no other solute molecules present. If finite solute concentrations lead to more favorable Boltzmann factors, then y < 1, and vice versa for less favorable interactions. Also, as Xq, 0, y 1, as expected. 

At infinite dilution the activity coefficients 7n and 7m are both unity, in accordance with equations (37.1) and (37.3), respectively, which define the standard states. In dilute solutions the values will also be approximately equal, but not necessarily unity because of failure to obey Henry s law. At appreciable concentrations, however, the molality of the solution is no longer proportional to the mole fraction of the solute, and even if the solution obeyed Henry s law 7m would not be unity. In such solutions 7n and 7m will be appreciably different. 

From the definition (Edward, 1964 Boyd, 1969 Yates and McClelland, 1974) of activity coefficients and of their standard states (infinite dilution in water), the log values of eqns (12)-(15) are proportional to the difference in standard chemical potentials in two media, water and the acid solution of interest. Log is, therefore,... 

Dissolution and condensation may be compared provided that the standard state of the solute is represented by (a) in the vapour state pressure 1 atm temperature T reference state ideal behaviour (b) in the liquid phase single solute in hypothetical liquid state at temperature T reference state infinitely diluted solution [21]. In these conditions the concentration unit is fugacity unit (atmosphere) in the vpaour phase and the mole fraction in the liquid phase. The phase equilibrium constant, K, is then replaced by another constant x given by the equation ... 

II The increment in the free energy, AF, in the reaction of forming the given substance in its standard state from its elements in their standard states. The standard states are for a gas, fugacity (approximately equal to the pressure) of 1 atm for a pure liquid or solid, the substance at a pressure of 1 atm for a substance in aqueous solution, the hyj)othetical solution of unit molahty, which has all the properties of the infinitely dilute solution except the property of concentration. 

Thus, with a Henry s law standard state, H° is the enthalpy in an infinitely dilute solution. For mixtures, in which we choose a Raoult s law standard state for the solvent and a Henry s law standard state for the solute, we can... 

Relative partial molar enthalpies can be used to calculate AH for various processes involving the mixing of solute, solvent, and solution. For example, Table 7.2 gives values for L and L2 for aqueous sulfuric acid solutions7 as a function of molality at 298.15 K. Also tabulated is A, the ratio of moles H2O to moles H2S(V We note from the table that L — L2 — 0 in the infinitely dilute solution. Thus, a Raoult s law standard state has been chosen for H20 and a Henry s law standard state is used for H2SO4. The value L2 = 95,281 Tmol-1 is the extrapolated relative partial molar enthalpy of pure H2SO4. It is the value for 77f- 77°. 

We showed in Section 7.3a that AMH - 0 for the change from the infinitely dilute solution to the standard state. 

AtH°4 takes the infinitely dilute solution to the Henry s law standard state. We have shown earlier that AH = 0 for this process. 

For a solution of a non-volatile substance (e.g. a solid) in a liquid the vapour pressure of the solute can be neglected. The reference state for such a substance is usually its very dilute solution—in the limiting case an infinitely dilute solution—which has identical properties with an ideal solution and is thus useful, especially for introducing activity coefficients (see Sections 1.1.4 and 1.3). The standard chemical potential of such a solute is defined as... 

In infinitely dilute solutions (in the standard state) ions do not interact, their electric field corresponds to that of point charges located at very large distances and the solution behaves ideally. As the solution becomes more concentrated, the ions approach one another, whence their fields become deformed. This process is connected with electrical work depending on the interactions of the ions. Differentiation of this quantity with respect to rc, permits calculation of the activity coefficient this differentiation is identical with the differentiation 3GE/5/iI and thus with the term RT In y,. 

If the fused salt does not exist at the temperature of interest, one normally uses the infinitely dilute solute standard state. While these equations can easily be converted to that basis, the results are not immediately useful for two reasons ... 

Thus AG° = -RT In/ , calculated at zero surface coverage, implies that the standard state for adsorbed S is pure solvent and for the adsorbate B is the ideal ats = 1, i.e., the pure solute with lateral interactions corresponding to an infinite distance between the molecules (i.e., the physical state corresponds to infinitely diluted solution of B). 

Solute. The standard state for the solute is the hypothetical unit mole fraction state (Fig. 16.2) or the hypothetical 1-molal solution (Fig. 16.4). In both cases, the standard state is obtained by extrapolation from the Henry s-Iaw line that describes behavior at infinite dilution. Thus, the partial molar enthalpy of the standard state is not that of the actual pure solute or the actual 1 -molal solution. 

To apply Equation (19.31) to experimental data, we must specify our choice of standard states, because the values of and of ahci depend on this choice. We shall use the hypothetical unit molality ratio standard state obtained by extrapolation from the infinitely dilute solution. By convention, m° is taken equal to 1 mol kg ... 

As the enthalpy of the dissolved sodium chloride in its standard state according to Henry s law is that of the infinitely dilute solution, A// , for the reaction in Equation (20.74) is... 

Consider a dilute ideal solution of the solute B (which could be gaseous, liquid, or solid at the temperature in question) in the solvent A. Suppose that more concentrated solutions do not behave ideally and, in particular, the state of pure liquid B cannot be attained by going to more and more concentrated solutions (e.g., by removing A by volatilization). It is possible to define a standard chemical potential pertaining to a hypothetical standard state of the ideal infinitely dilute solution as the limit ... 

Conceptually, although these standard states are not infinite dilution states, they reflect, through the linear extrapolation, the properties of the infinitely dilute solutions (61 Mil). 

In this equation the standard state corresponds to the state that results from letting fw - 1 and xw - 1, in which case = n°s w. Letting/ - 1 is equivalent to saying that the surfactant behaves ideally, and letting xw - 1 is equivalent to having pure surfactant possessing the kind of interactions it has when surrounded by water. Physically, this corresponds to an infinitely dilute solution of surfactant in water. Using the primed symbol to represent the chemical potential of surfactant in micelles per mole of micelles, we write... 

The definition of the standard state contains subtleties beyond the scope of this book. For Reaction 6-3, the standard state of H+ or Cl- is the hypothetical state in which each ion is present at a concentration of 1 M but behaves as if it were in an infinitely dilute solution. That is, the standard concentration is 1 M, but the standard behavior is what would be observed in a very dilute solution in which each ion is unaffected by surrounding ions. 

All species are aqueous unless otherwise indicated. The reference state for amalgams is an infinitely dilute solution of the element in Hg. The temperature coefficient, dE°/dT, allows us to calculate the standard potential, E°(T), at temperature T E°(T) — Ec + (dE°/dT)AT. where A T is T — 298.15 K. Note the units mVIK for dE°ldT. Once you know E° for a net cell reaction at temperature T, you can find the equilibrium constant, K, for the reaction from the formula K — lOnFE°,RTln w, where n is the number of electrons in each half-reaction, F is the Faraday constant, and R is the gas constant. 

As we described earlier, the calorimetric determination of log K allows one to also get ArH for reaction (15.37). The values reported by Izatt and his colleagues were obtained in an aqueous solution with an ionic strength of 0.1. Izatt reports that enough measurements were made in more dilute solutions to show that the enthalpy of dilution to the infinitely dilute solution (the standard state) is small and can be ignored. Hence, we will assume that the ArH values reported are the standard state ATH° values. Thus we have available, ArG°, obtained from equation (15.42), and ArS ° obtained from equation (15.43). 

In equations (18.91) and (18.92), C° 2 and V are the partial molar heat capacity and partial molar volume of the surfactant in the infinitely dilute solution (standard state values). 

The physical state of each substance is indicated in the column headed State as crystalline solid (c), liquid (liq), gaseous (g), or amorphous (amorp). Solutions in water are listed as aqueous (aq). Solutions in water are designated as aqueous, and the concentration of the solution is expressed in terms of the number of moles of solvent associated with 1 mol of the solute. If no concentration is indicated, the solution is assumed to be dilute. The standard state for a solute in aqueous solution is taken as the hypothetical ideal solution of unit molality (indicated as std state, m = 1). In this state the partial molal enthalpy and the heat capacity of the solute are the same as in the infinitely dilute real solution (aq. m). 

When the infinitely dilute solution, with respect to all solutes, is used as the reference state of the solution at all temperatures and pressures, Ap c approaches zero as all cfs approach zero. Thus, the standard state of the solvent is the pure solvent at all temperature and pressures and is identical to the reference state of the solvent for all thermodynamic functions. 

The definition is completed by assigning a value to m and (f>c in some reference state. To conform with the definitions made in Sections 8.9 and 8.10, the infinitely dilute solution with respect to all molalities or molarities is usually used as the reference state at all temperatures and pressures, and both m and c are made to approach unity as the sum of the molalities or molarities of the solutes approaches zero. The standard state of the solvent is again the pure solvent, and is identical to its reference state in all of its thermodynamic functions. 

The problems of the reference and standard states are slightly different when we use the infinitely dilute solution-of all solutes as the reference states. When we take the reference state of both the component and the species as the infinitely dilute solution, Equation (8.168) becomes... 

The reference state of the electrolyte can now be defined in terms of thii equation. We use the infinitely dilute solution of the component in the solvent and let the mean activity coefficient go to unity as the molality or mean molality goes to zero. This definition fixes the standard state of the solute on the basis of Equation (8.184). We find later in this section that it is neither profitable nor convenient to express the chemical potential of the component in terms of its molality and activity. Moreover, we are not able to separate the individual quantities, and /i . Consequently, we arbitrarily define the standard chemical potential of the component by... 

Special consideration must be given to systems involving liquid solutions of at least one solid component, for which the choice of either the pure solid or pure supercooled liquid as the standard state is not convenient. This case is encountered for all solutions in which the pure solute is not chosen as the reference state. As an example, we consider an aqueous solution of a solid B and choose the reference state to be the infinitely dilute solution. Then a general change of state for the formation of the solution from the components is written as... 

When the reference state is the infinitely dilute solution, the standard state for the enthalpy is also the infinitely dilute solution. We then change the standard state of component B from the pure solid to the infinitely dilute solution by adding to and subtracting from Equation (9.30) the quantity n2H2, where H2 is the partial molar enthalpy of the component in the... 

Many tables of values of standard changes of enthalpy of formation list values for individual ions, particularly in aqueous solutions. In order to do so, an arbitrary definition must be introduced because the properties of individual ions in solution cannot be determined. We consider an electrolyte Mv+Av which is completely ionized in the infinitely dilute solution. We choose this solution to be the standard state for the enthalpy. The enthalpy of this solution per mole of solute, H, is given by... 

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